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arXiv · 1907.04175

A method for computing the Perron root for primitive matrices

Doulaye Dembélé
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Abstract

Following the Perron-Frobenius theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix $A$, there is a positive rank one matrix $X$ such that $B = A \circ X$, where $\circ$ denotes the Hadamard product of matrices, and such that the row (column) sums of matrix $B$ are the same and equal to the Perron root. An iterative algorithm is presented to obtain matrix $B$ without an explicit knowledge of $X$. The convergence rate of this algorithm is similar to that of the power method but it uses less computational load. A byproduct of the proposed algorithm is a new method for calculating the first eigenvector.

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Doulaye Dembélé. 2019-07-09. A method for computing the Perron root for primitive matrices. https://arxiv.org/abs/1907.04175

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